reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  ( for n being Nat holds seq.n <> 0.X &
   Rseq.n = ||.seq.(n+1).||/||.seq.n.|| ) &
  Rseq is convergent & lim Rseq < 1 implies seq is absolutely_summable
proof
  assume that
A1: for n being Nat
    holds seq.n <> 09(X) & Rseq.n = ||.seq.(n+1).||/||.seq.n.|| and
A2: Rseq is convergent & lim Rseq < 1;
  for n being Nat
  holds ||.seq.||.n > 0 & Rseq.n = ||.seq.||.(n+1)/||.seq.||.n
  proof
    let n be Nat;
A3: ||.seq.n.|| <> 0 by A1,BHSP_1:26;
    ||.seq.n.|| >= 0 by BHSP_1:28;
    hence ||.seq.||.n > 0 by A3,BHSP_2:def 3;
    Rseq.n = ||.seq.(n+1).||/||.seq.n.|| by A1
      .= ||.seq.||.(n+1)/||.seq.n.|| by BHSP_2:def 3;
    hence Rseq.n = ||.seq.||.(n+1)/||.seq.||.n by BHSP_2:def 3;
  end;
  hence thesis by A2,SERIES_1:26;
end;
